complete the identity:
sec x - 1/sec x = ?
sec x - 1/sec x
= 1/cosx - cosx
= (1 - cos^2x)/cosx
= sin^2x/cosx
= (sinx/cosx)sinx
= tanxsinx
To complete the identity, we need to find a common denominator and then combine the terms.
The common denominator for both terms is sec(x). So, we can convert 1 to sec(x) by multiplying it by sec(x)/sec(x):
sec(x) - 1/sec(x) = sec(x) - sec(x)/sec(x)
Now, we can combine the terms:
sec(x) - sec(x)/sec(x) = (sec(x) * sec(x) - sec(x))/sec(x)
Using the distributive property, we have:
(sec(x) * sec(x) - sec(x))/sec(x) = (sec^2(x) - sec(x))/sec(x)
Thus, the completed identity is:
sec(x) - 1/sec(x) = (sec^2(x) - sec(x))/sec(x)
To complete the given identity, sec x - 1/sec x, we can start by finding a common denominator for the two terms. The common denominator would be sec x.
Now, let's rewrite the expression using the common denominator:
(sec^2 x - 1) / sec x
The expression (sec^2 x - 1) is a well-known identity for secant squared: sec^2 x = tan^2 x +1.
Substituting this identity into the expression, we have:
(tan^2 x + 1 - 1) / sec x
Simplifying further, we find:
tan^2 x / sec x
tan x * sin x / (1/cos x)
tan x * sin x * cos x
tan x * sin x * cos x can be further simplified using the identity sin x * cos x = 1/2 * sin(2x), so we have:
tan x * 1/2 * sin(2x)
(1/2) * tan x * sin(2x)
Therefore, the simplified form of sec x - 1/sec x is:
(1/2) * tan x * sin(2x)